GIFT   OF 


Professor  Montgomery's  Discoveries 
in  Celestial  Mechanics 


BY 


L.  A.  REDMAN 


Pernau-Walsh  Printing  Co. 

753  Market  Street 

San  Francisco 


Copyright,  1919 

by 
L.  A.  REDMAN 

Qift  of 


-•ere  e  »-•»  ••»••••«  »« 


"Thus  the  Seer, 
With  vision  clear, 
Sees  forms  appear  and  disappear, 
In  the  perpetual  round  of  strange, 
Mysterious  change 

From  birth  to  death,  from  death  to  birth, 
From  earth  to  heaven,  from  heaven  to  earth; 
Till  glimpses  more  sublime 
Of  things,  unseen  before, 
Unto  his  wondering  eyes  reveal 
The  Universe  as  an  immeasurable  wheel 
Turning  for  evermore 
In  the  rapid  and  rushing  river  of  Time." 

Longfellow. 


The  views  herein  set  forth  were  made  known  to  me  in  the  year 
1886  by  the  late  John  J.  Montgomery,  afterwards  a  professor  in 
Santa  Clara  College,  and  at  that  time  the  model  referred  to  in  the 
text  was  exhibited  to  me  and  it  is  now  in  my  possession.  On  sev- 
eral occasions  thereafter  I  urged  him  to  publish  his  discovery,  the 
last  time  in  the  year  1909,  about  two  years  before  his  death.  He 
did  not  do  so  however,  and  on  my  pressing  him  for  an  explana- 
tion on  the  last  occasion,  he  said  that  his  failure  to  publish  was 
not  because  of  any  doubt  as  to  the  soundness  of  his  views,  but 
that  the  discovery  made  by  him  was  connected  with  or  inci- 
dental to  certain  underlying  principles,  and  that  in  due  time  he 
would  reveal  it  in  its  relation  to  such  principles.  I  do  not 
know  what  he  had  in  mind.  Possibly  principles  of  molecular 
action,  or  it  may  be  that  he  saw  and  intended  presently  to 
explain  how  the  progressive  changes  which  he  proved  were 
mechanically  necessitated  were  incidents  of  the  dissipation  of 
energy  in  the  solar  system.  This  (to  a  limited  extent)  was  per- 
ceived by  Professor  Percival  Lowell,  and  was  expounded  in  his 
"The  Evolution  of  Worlds",  which  was  published  at  about  the 
time  I  last  saw  Montgomery.  Professor  Lowell  says  that  Sir  Robert 
Ball  was  the  first  to  suggest  the  "argument"  which  he  employed 
(p.  145).  I  do  not  recall  that  Montgomery  ever  mentioned  to  me 
this  phase  of  the  subject.  I  first  became  acquainted  with  this 
principle  on  reading  Lowell  last  year  when  I  began  my  investiga- 
tion. Montgomery  met  with  a  sudden  death  by  the  fall  of  an  air- 
plane with  which  he  was  experimenting.  During  his  lifetime  (in 
1905)  there  appeared  in  the  newspapers  an  article,  telegraphed 
from  Cambridge,  Massachusetts,  setting  forth  that  Professor  Wil- 
liam H.  Pickering  had  discovered  that  the  planets  were  "keeling 
over",  and  thereupon,  without  Montgomery's  knowledge,  I  sent  to 
the  San  Francisco  "Bulletin"  the  following  communication  which 
it  published: 

"CALIFORNIA  ENTERTAINING  A  GENIUS  UNAWARES. 

San  Francisco,  March  27,  1905. 
Editor  Bulletin: 

Referring  to  the  dispatches  in  The  Bulletin  of  the  15th  inst. 
from  Cambridge,  Massachusetts,  and  the  Lick  Observatory, 
relating  to  Professor  Pickering's  theory  that  'all  the  planets 
have  keeled  over  or  are  keeling  over',  I  feel  impelled  to  make 
public  the  fact  that  this  discovery  was  made  by  Mr.  John  J. 


Montgomery,  now  a  professor  in  Santa  Clara  College,  more 
than  twenty  years  ago.  It  was  disclosed  by  him  to  me  in  1886 
or  1887.  With  him  it  was  more  than  a  theory.  By  means  of 
a  mechanical  device — a  sort  of  orrery — he  physically  demon- 
strated that  spheres  having  a  retrograde  rotation  being  in 
unstable  equilibrium,  will  necessarily  'keel  over'.  The  parts 
nearest  the  sun  of  a  planet  having  a  retrograde  rotation  move 
with  relation  to  the  sun  more  rapidly  than  those  parts  on  its 
remote  side ;  hence  the  former  tend  to  move  away  from  the 
sun  while  the  latter  have  a  tendency  to  move  toward  it.  The 
result  is  that  the  planet  turns  over  and  establishes  a  condition 
of  stable  equilibrium — one  in  which  the  inner  parts  tend  to 
move  towards  the  sun  and  the  outer  parts  away  from  it.  I 
have  not  sufficient  knowledge  of  the  subject  to  set  forth  accu- 
rately all  the  remarkable  implications  of  this  great  discovery 
— some  of  which  I  believe  are  still  occupying  Mr.  Montgom- 
ery's attention.  I  may  say,  however,  that  it  accounts  for  the 
precession  of  the  equinoxes,  the  tides,  the  transfixed  position 
of  the  moons  of  Jupiter  and  the  earth  and  of  Venus  ( ? ) ,  the 
half-tilted  over  condition  of  Uranus,  the  retrograde  rotation 
of  Neptune  and  the  retrograde  translatory  motion  of  Saturn's 
ninth  moon. 

In  my  judgment  this  brilliant  revelation  eclipses  any  dis- 
covery made  in  astronomical  science  since  the  days  of  Kepler 
and  Newton.  In  unassuming  John  Montgomery,  whose  recent 
experiments  in  wireless  telegraphy  and  aerial  navigation  have 
been  attracting  public  attention,  California  has  these  many 
years  been  ' entertaining  an  angel  unawares.'  His  self-abnega- 
tion is  so  extreme  that  unless  some  one  else  drew  attention  to 
his  claims  the  merit  of  the  discovery  referred  to  would  nat- 
urally be  bestowed  wholly  upon  Professor  Pickering.  I  do  not 
question  the  latter 's  claim  to  originality,  nor  do  I  know  when 
the  discovery  was  made  by  him,  but  I  do  know  that  it  was 
made  many  years  ago  by  Mr.  Montgomery. 

Respectfully  yours, 

L.  A.  REDMAN." 


I  learned  upon  investigating  the  subject  last  year  that  Professor 
Pickering  had  first  advanced  his  theory  in  1893.  But  the  reading 
of  his  articles  disclosed  that,  while  he  contended  that  polar 
inversion,  where  the  rotation  is  retrograde,  must  ensue,  his  explan- 
ation of  the  cause  thereof  is  totally  different  from  that  of 
Montgomery  and  is,  as  I  think  I  have  clearly  shown,  erroneous. 

Montgomery  was  a  firm  believer  in  the  nebular  hypothesis.  He 
also  believed,  as  Kirkwood  did,  ("The  American  Journal  of 
Science",  Vol.  36,  p.  1),  and  as  Professor  Pickering  and  others  do, 
that  the  planets  formed  from  detached  parts  of  the  nebula  as  it 


contracted  would  have  retrograde  rotations.  I  have  not  however 
dealt  with  this  feature  of  the  problem.  Postulating  this,  I 
have  set  forth  the  argument  showing  how  the  planets  are  inverted, 
and  then  how  forward  rotation  (which  inversion  involves)  is 
retarded  until  the  rotation  and  revolution  periods  coincide;  and 
finally  how  the  planets  and  their  satellites  will  be  drawn  into 
the  sun.  I  have  also  shown  the  harmony  between  Montgomery's 
discovery  and  the  proposition  that  energy  will  be  dissipated  until 
it  is  reduced  to  the  least  quantity  consistent  with  the  original 
and  unchanging  moment  of  momentum  of  the  system.  And  I 
have  pointed  out  (what  Ball  and  Lowell  did  not  see)  that  their 
" argument"  involves  the  proposition  that  by  dissipation  of  energy 
the  matter  composing  the  nebula  will  ultimately  be  concentrated 
in  a  cold  rotating  body. 

There  is  one  feature  of  the  process  which  I  have  merely  indi- 
cated without  going  into  it  in  detail,  namely,  the  inversion  of 
the  orbits  of  the  satellites.  The  planets  do  not  "carry  over" 
their  moons,  but  the  orbits  of  the  moons  are  "carried  over" 
by  the  sun.  And  where  the  differential  attraction  of  a  satellite  is 
greater  than  that  of  the  sun  it  controls  the  inversion  of  its  prim- 
ary. In  the  case  of  the  few  very  remote  satellites  which  still  have  a 
retrograde  revolution,  the  inversion  force  has  been  so  feeble  that 
little  progress  has  been  made.  And  in  this  connection  it  is  to  be 
noted  that  while  Mercury  and  Venus  (which  have  no  moons)  turn 
always  the  same  face  to  the  sun,  the  earth  will  not  do  so  until 
after  it  has  absorbed  the  moon.  It  is  the  moon  and  not  the  sun 
which  controls  the  earth's  inversion,  and  it  will  also  control  the 
earth's  rotation  as  its  differential  attraction  is  greater  than  that  of 
the  sun.  Hence,  the  earth's  rotation  will  only  be  retarded  until 
the  day  and  month  are  the  same.  The  earth  and  moon  will  finally 
rotate  and  revolve  as  if  united  by  a  bar.  Thereupon  the  moon  will 
spiral  inward  upon  the  earth;  and  finally  the  earth-moon  will 
spiral  into  the  sun. 

In  December,  1917,  I  asked  Professor  Campbell,  Director  of  the 
Lick  Observatory,  if  the  views  of  Montgomery  were  known  to  and 
accepted  by  astronomers  as  correct,  and  was  informed  by  him  that 
the  theory  of  Professor  Pickering  was  well  known,  but  that  it  was 
not  generally  accepted.  I  thereupon  made  an  investigation  and 
was  amazed  to  find  that  not  only  was  there  a  want  of  agreement 
amongst  those  who  had  given  the  matter  consideration,  but  that 
the  arguments  made  in  support  of  the  propositions  advanced  by 
the  various  writers  on  the  subject  were  of  the  loosest  character, 


and  that  some  of  the  positions  taken  were  manifestly  untenable. 
Everywhere  I  found  confusion  and  contradiction,  and  nowhere 
any  recognition  of  the  controlling  factor  detected  by  Montgomery. 
I  found  Professors  Pickering  and  Moulton  differing  upon  what  the 
former  denominated  an  ' '  elementary ' '  mechanical  proposition ;  and 
I  also  found  that  men  eminent  in  mathematical  research  had 
reached  conclusions  which,  however  profound  the  mathematical 
reasoning  by  which  they  were  deduced,  were  obviously  erroneous. 

In  a  recent  number  of  "Popular  Astronomy"  Mr.  Cordeiro 
says  that  the  mention  of  Sir  George  Darwin  as  an  "authority" 
makes  him  "shudder".  Mr.  Cordeiro,  however,  falls  into  the 
same  error  as  Professor  Pickering.  He  attributes  inversion  to 
tidal  action  which  he  says  is  always  "positive".  It  is  not 
positive  when  the  rotation  is  forward. 

Another  remarkable  thing  is  that  no  one  seems  to  have  perceived 
that  there  are  no  "couples"  in  celestial  mechanics.  Bodies  in  space 
caused  to  rotate  by  attraction  do  not  turn  on  fixed  axes,  but  every 
particle  of  an  attracted  body,  notwithstanding  such  rotation,  moves 
towards  the  attracting  body.  The  failure  to  note  this  accounts 
for  the  error  into  which  Professor  Newcomb  fell  in  his  analysis 
of  the  cause  of  the  identity  of  the  periods  of  the  moon's  revolution 
and  rotation.  It  probably  also  accounts  for  Professor  Moulton 's 
admitted  inability  to  understand  the  reason  for  such  identity 
("An  Introduction  to  Astronomy",  1916,  p.  458). 

This  strange  condition  of  affairs  presents  an  interesting 
subject  for  study.  It  is  probably  due  to  a  habit  of  mind 
produced  by  a  too  exclusive  mathematical  discipline.  It  has 
been  observed  that  mathematicians  are  prone  to  accept  as 
indisputably  correct  conclusions  deduced  mathematically  but 
resting  upon  erroneous  or  inaccurate  data,  notwithstanding 
that  they  are  manifestly  unsound.  "The  mathematician",  says 
Herbert  Spencer,  "in  dealing  with  contingent  matters  does  not  go 
wrong  in  reasoning  from  his  premises ;  he  goes  wrong  in  his  choice 
of  premises.  He  continually  assumes  that  these  are  simple  when 
they  are  really  complex — omits  some  of  the  factors.  His  habit  of 
thought  is  that  of  dealing  with  few  and  quite  definite  data  and  he 
carries  that  habit  of  thought  into  regions  where  the  data  are  many 
and  indefinite,  and  proceeds  to  treat  a  few  of  them  as  though  they 
were  all,  and  regards  them  as  definite."  And  it  may  be  further 
observed  that  even  where  the  premises  or  "starting  points"  are 
correct  the  mathematician  is  apt  to  overlook  changes  in  some  of 
the  factors  produced  by  subsequent  occurrences.  But  whatever 

8 


the  explanation  may  be,  the  extraordinary  situation  is  presented 
of  mathematicians  studying  for  generations  the  mechanical  prob- 
lems herein  considered  without  solving  them,  and  profoundly  con- 
fusing the  subject  by  applying  to  them  mathematical  principles 
and  formulae  applicable  to  ideal  but  not  real  conditions.  Astro- 
nomical growth  has  been  stunted  by  the  procrustean  assumptions 
of  the  mathematicians. 

Aside  from  Montgomery,  Ball  and  Lowell  came  nearest  to  solv- 
ing the  problems  in  question  by  applying,  as  I  have  said,  the 
principle  of  the  dissipation  of  energy.  But,  as  above  stated, 
neither  of  them  saw  the  most  important  implication  of  this 
doctrine — the  absorption  by  the  sun  of  the  planets  and  their  satel- 
lites— and  neither  of  them  mastered  the  subject  as  a  mechanical 
problem,  and  consequently  both  fell  into  serious  errors.  No  one 
but  Montgomery  perceived  the  pregnant  proposition  that  the  retro- 
grade-rotating parts  of  a  planet  are  virtually  nearer  the  point  of 
attraction  than  the  relatively  more  rapidly  moving  forward  parts. 
Hence  a  retrograde-rotating  planet  is  in  unstable  equilibrium  and 
will  invert  under  the  influence  of  attraction.  It  is  of  course  true, 
as  Lowell  observes,  that  there  is  more  "superfluous  energy"  in  a 
retrograde  than  in  a  forward-rotating  body,  and  that  by  dissipa- 
tion of  energy,  inversion  (which  converts  retrograde  into  forward 
rotation)  will  ensue  (p.  145).  But  he  did  not  see  how  this  was 
mechanically  brought  about.  He  erroneously  asserts  that  "tidal 
action"  will  "accomplish  the  end".  He  strangely,  however,  does 
not  refer  to  the  ' '  annual  tide ' '  theory  of  Professor  Pickering.  Nor 
does  he  allude  to  Professor  Moulton's  attempted  refutation  of  the 
tidal  theory.  Professor  Lowell  is  also  in  error  in  asserting  that 
tidal  friction  acts  as  a  "brake"  on  rotation;  and  further  in 
error  in  contending  that  the  tidal  bulges  "carry  over"  the 
orbits  of  the  planets'  satellites.  He  did  not  see  that  where  the 
differential  attraction  of  a  moon  is  greater  than  that  of  the  sun, 
the  inversion  and  rotation  of  the  planet  will  be  controlled  by  the 
moon  and  not  by  the  sun.  In  such  case,  as  I  have  already  indi- 
cated, the  sun  will  invert  the  orbit  of  the  moon  and  the  moon's 
attraction,  assisted  by  that  of  the  sun,  will  invert  the  planet. 

Professor  Moulton's  attack  on  Professor  Pickering's  "annual 
tide"  theory  is  as  ill-based  as  the  theory  itself.  He  fell  into  the 
error  of  applying  to  a  spheroid  with  tides,  principles  applicable 
only  to  rigid  spheres.  He  impliedly  denied  precessional  motion. 


He  said  that  a  planet  would  not  tilt,  but  that  the  axis  (in  a 
spheroid)  would  shift! 

Ball  and  Lowell  contend  that  the  obliquity  of  the  earth's  axis  is 
decreasing  and  that  it  will  continue  to  decrease  until  it  is  perpen- 
dicular to  the  orbital  plane;  but  Ball  failed  to  explain  how  this 
was  mechanically  caused,  and  Lowell,  as  I  have  said,  erroneously 
attributed  the  result  to  "tidal  action"  (which  Darwin  says  is 
causing  increasing  obliquity).  And  they  both  impliedly  denied 
the  true  cause,  namely  the  attraction  of  the  moon  and  sun  on  the 
earth's  equatorial  protuberance.  They  assert  that  such  attrac- 
tion causes  precession  exclusively,  whereas  precession  would  not 
occur  unless  the  axis  were  tilted.  Rotation  does  not  prevent  tilt- 
ing, but  tilting  proceeds  accompanied  by  precession.  The  two 
motions  compounded  cause  the  poles  to  move  in  spirals. 

On  completing  my  investigation  I  sent  to  Professor  Camp- 
bell (last  July)  a  communication  wherein  I  set  forth  as  the 
result  of  refreshed  and  cleared-up  recollection  of  Montgomery's 
views  and  my  own  study  of  the  subject,  the  principles  which 
govern  the  motions  of  the  planets  and  their  satellites;  and  from 
time  to  time  thereafter  sent  him  additional  communications  in 
which  various  aspects  of  the  subject  were  dealt  with  and  the  con- 
tentions and  arguments  of  Pickering,  Moulton,  Darwin,  and 
others,  reviewed.  These  are  reproduced  here  together  with  two 
previous  letters  wherein  I  stated  Montgomery's  views  in  a 
general  way. 

As  excuses  for  such  an  informal  presentation  of  a  matter  of  so 
great  importance,  I  plead  want  of  time  and  equipment  to  develop 
and  present  a  better  organized  and  complete  discussion  of  the 
subject,  as  well  as  a  rather  sharp  desire  to  give  the  discovery  wide 
publicity  without  further  delay.  I  have  no  doubt  as  to  the  verdict 
that  will  be  rendered.  The  name  of  John  J.  Montgomery  will  be 
written  high  up  in  the  hall  of  scientific  fame — beside  that  of  New- 
ton, Charles  Darwin,  and  other  great  and  original  thinkers. 


San  Francisco,  May  6,  1919. 


10 


Letters  to  Professor  Campbell  of  December  21,  1917,  and 

January  23,  1918,  Setting  Forth  the  Views 

of  Professor  Montgomery. 

Many  years  ago  the  late  Professor  John  J.  Montgomery  of 
Santa  Clara  College  informed  me  that  the  reason  why  the  moon 
rotated  on  its  axis  in  the  same  period  that  it  takes  to  go 
round  the  earth  is  that  such  a  position  is  one  of  stable  equilibrium 
— one  into  which  the  moon  would  necessarily  pass  no  matter  how 
rapidly  or  in  what  direction  it  was  rotated.  If,  for  instance,  it  was, 
given  a  retrograde  rotation  it  would  in  course  of  time  turn  over, 
thereby  assuming  a  forward  rotation,  and  that  its  rotary  motion 
would  be  gradually  retarded  until  it  finally  reached  its  present 
state  of  stable  equilibrium.  He  showed  me  the  reason  for  this 
and  proved  it  by  a  model  which  he  still  had  when  I  last  saw  him, 
about  two  years  before  his  death.  Professor  Montgomery  informed 
me  that  in  his  opinion  the  planets  originally  all  had  retrograde 
rotations  and  that  they  have  all,  or  nearly  all,  been  capsized  by 
attraction.  Their  ultimate  state,  he  said,  would  correspond  to  that 
of  our  moon,  namely,  each  of  them  having  a  rotation  period  equal 
to  its  revolution  period. 

I  have  for  a  long  time  been  curious  to  know  whether  or  not 
these  views  are  known  to  astronomers,  and  if  so,  to  what  extent, 
if  any,  they  have  been  accepted.  Evidently  Professor  Larkin  either 
never  heard  of  them  or  takes  no  stock  in  them,  as  the  enclosed 
clipping  shows.  [Referring  to  a  statement  that  the  period  of  the 
moon's  revolution  and  rotation  has  been  the  same  ever  since  the 
moon  "fell  into  its  orbit  as  it  now  stands".]  But  clearly  the 
chances  are  infinity  to  one  against  an  accidental  exact  correspond- 
ence between  the  time  of  the  moon's  rotation  on  its  axis  and  the 
time  of  its  revolution  round  the  earth.  I  understand  that  the  same 
thing  is  true  of  Jupiter's  moons,  and  if  so,  it  passes  comprehension 
how  any  one  can  believe  that  such  a  thing  could  have  resulted 
otherwise  than  from  some  common  cause. 


As  heretofore  stated,  Professor  Montgomery  contended  that  a 
central  body  will  necessarily  bring  a  rotating  body  circling  round 
it  into  the  relation  which  the  moon  bears  to  the  earth.  If  the 
planet  or  satellite  have  a  retrograde  rotation,  the  central  attracting 

11 


force  will  first  capsize  it — which  movement  is  along  the  line  of 
least  resistance — and  then  slow  down  its  axial  speed  until  it  makes 
but  one  rotation  in  its  course  round  the  central  body;  and  further 
that  if  such  planet  or  satellite  have  no  rotation,  the  central 
force  will  initiate  such  rotation  and  gradually  increase  its  speed 
until  axial  and  annual  correspondence  is  attained.  The  parts 
of  a  planet  which  has  no  rotation  will  not  move  in  a  circle — 
a  condition  which  the  central  force  will  not  tolerate.  A  correc- 
tion of  it  results  in  turning  the  planet  on  its  axis  until  such 
an  axial  speed  is  attained  that  all  of  its  parts  move  round  the 
central  body  in  a  circle;  that  is  to  say,  with  the  same  side  always 
towards  the  center.  And  if  the  rotation  be  greater  (as  in  the 
case  of  the  earth)  the  parts,  instead  of  moving  in  circles,  will 
move  in  wave  lines — another  unstable  condition  which  the  central 
attracting  force  will  correct. 

As  above  stated,  Professor  Montgomery  contended  that  a  planet 
with  a  retrograde  rotation  will  gradually  capsize,  the  reason  being 
that  the  inner  parts  move,  in  relation  to  the  central  ~body,  faster 
than  the  outer  parts.  If,  for  instance,  the  rotation  were  equal  to 
the  speed  of  the  planet  through  space,  the  outer  parts,  in  rela- 
tion to  the  central  body,  would  be  motionless,  and  if  detached  from 
the  planet  of  which  they  form  a  part,  would  fly  straight  to  the 
central  body  (assuming,  of  course,  the  removal  of  the  other  parts 
of  the  planet).  The  inner  parts,  on  the  other  hand,  if  detached, 
would  move  away  from  the  central  body  in  a  larger  orbit  than  that 
of  the  planet  as  a  whole.  The  effect  of  these  two  opposing  forces 
is  to  turn  the  planet  over  gradually.  And  Professor  Montgomery 
proved  this  by  the  device  to  which  I  referred  in  my  former  letter. 
Of  course,  the  natural  and  artificial  conditions  are  not  precisely 
the  same,  but  their  points  of  difference  are  immaterial.  The 
attractive  force  upon  all  of  the  parts  is  simply  concentrated  at 
the  center  of  the  wheel.  The  effect  of  the  tug  at  the  center 
is  the  same  as  the  force  of  attraction  distributed  throughout 
the  entire  body.  The  outer  parts  having  a  slower  motion  with 
respect  to  the  central  force,  due  to  retrograde  rotation,  will  be 
more  affected  thereby  than  the  relatively  rapidly  moving  inner 
parts.  They  will  tend  to  go  further  towards  the  central  body  in  a 
given  time  than  the  inner  parts.  The  situation  is  the  same  as  if 
the  inner  and  outer  parts  were  disconnected.  Place  two  rifles  one 
above  the  other  and  fire  them,  the  lower  one  discharging  its  bullet 
with  greater  force  than  the  upper  one.  Gravity  will  cause  the 
upper  bullet  to  cross  the  course  taken  by  the  more  rapidly  moving 


12 


lower  one.  And  if  they  be  connected  as  in  the  case  of  a  planet 
a  capsizing  movement  will  result.  And  the  capsizing  will  continue 
until  the  body  rotates  on  an  axis  at  right  angles  to  the  plane  of 
its  orbit.  It  is  still,  however,  in  unstable  equilibrium,  which,  as 
above  stated,  consists  in  the  parts  moving  in  wave  lines  instead 
of  in  smooth  circles  round  the  central  body. 

Professor  Montgomery  believed  that  all  of  the  planets  orig- 
inally had  retrograde  rotations  and  he  informed  me  that  this 
belief  was  confirmed  by  experiments  made  by  him.  He  said  that 
they  are  all  in  course  of  inversion  except  Venus  and  Mercury, 
where  the  process  has  been  completed.  The  force  which  produces 
this  result  is  proportional  to  the  times  of  revolution  thereby  ac- 
counting for  the  condition  of  the  innermost  planets  and  for  the 
fact  that  Uranus  is  but  half  turned  over  and  that  Neptune 
still  has  a  retrograde  rotation. 

I  realize  that  I  am  writing  in  a  more  confident  tone  than  my 
knowledge  or  capacity  in  such  matters  justifies,  but  I  cannot  help 
being  deeply  impressed  by  the  apparent  truth  of  these  theories 
of  Professor  Montgomery,  which  I  am  anxious  to  have  confirmed 
and  made  known  if  sound,  or  if  unsound,  to  see  refuted. 


Letter  of  July  24,  1918,  After 
Investigating  the  Subject. 

Professor  Montgomery  dissented  from  the  views  of  those  who 
say  that  in  the  absence  of  tides  there  will  be  no  inversion  or 
retardation  of  rotation.  He  contended  that  there  would  neces- 
sarily be  inversion  and  retardation  in  the  case  of  any  real 
body,  regardless  of  tides.  A  spheroid  would  invert,  he  claimed, 
if  it  rotated  retrograde-wise,  and  ultimately  would  be  brought 
into  a  condition  where  the  rotation  and  revolution  periods 
would  be  the  same.  The  same  thing  is  true  with  respect  to  any 
body  except  a  perfect  (ideal)  sphere.  An  iron  bar  rotating  in 
space  will  finally  point  one  end  towards  the  central  attracting 
body  as  it  rotates  and  revolves  around  it,  and  a  non-rotating  bar 
will  be  caused  to  rotate  by  attraction. 

While  Kirkwood  (in  1864)  contended,  as  did  Montgomery,  that 
the  planets  originally  had  retrograde  rotations,  the  process  by 
which  forward  rotation  was  brought  about,  as  described  by  him, 
is  essentially  different  from  that  asserted  by  Montgomery.  Kirk- 
wood's  contention  was  that  tidal  friction  retarded  retrograde 

13 


rotation  until  the  day  and  year  corresponded,  and  that  thereupon 
forward  rotation  was  initiated.  He  failed  to  note  the  fact  that,  as 
contended  by  Montgomery,  attraction  will  invert  a  retrograde- 
rotating  planet.  As  I  stated  to  you  in  a  previous  letter,  Montgom- 
ery in  1886  first  informed  me  regarding  his  views  and  experiments, 
and  he  also  said  that  in  respect  to  the  capsizing  and  retarding 
processes  they  were  original. 

At  first  (June,  1893)  Professor  Pickering  advanced  the  theory 
that  inversion  of  a  retrograde-rotating  body  would  be  caused  by 
"annual  tides"  produced  by  attraction.  He  said  that  a  planet 
with  a  retrograde  rotation  was  in  unstable  equilibrium  and  that 
such  tides  would  upset  it.  A  few  months  later,  however,  after 
acknowledging  the  inadequacy  of  his  tidal  theory  (September, 
1893,  "Astronomy  and  Astro-physics",  Vol.  12,  p.  692),  he  said 
that  "a  second  cause  of  inversion  has  recently  been  found  which 
in  its  action  is  probably  still  more  important  than  that  of  the 
tides",  and  argued  that  a  rigid  body,  such  as  a  wheel  for  ex- 
ample, would  be  upset  by  gyroscopic  action.  But  this  "second 
cause"  is  as  inadequate  as  the  "annual  tide".  When  the  rotation 
of  the  wheel  becomes  forward  and  the  rear  rim  is  nearer  the 
point  of  attraction  than  the  front  rim,  the  axis,  on  Professor 
Pickering's  theory,  would  be  precessionally  tilted  into  greater 
obliquity.  Perhaps  his  perception  of  this  accounts  for  the  fact 
that  he  has  not  in  any  of  his  subsequent  articles  referred  to 
this  "second  cause".  Apparently  for  this  reason  he  has  aban- 
doned it,  but  he  has  not,  so  far  as  I  am  aware,  expressly 
withdrawn  his  admission  of  inadequacy  with  respect  to 
his  "annual  tide"  theory.  It  is  plain  that  he  does  not 
understand  how  inversion  is  brought  about.  He  fails  to 
note  the  pivotal  fact  which  I  think  no  one  but  Montgom- 
ery perceived,  that  the  retrograde,  and  therefore,  relatively  to  the 
attracting  body,  slowly  moving  outer  parts,  are  drawn  inward  by 
attraction,  and  the  inner  parts,  moving  with  relatively  increased 
velocity,  tend  to  go  outward.  In  the  case  of  Uranus,  for  example, 
attraction  directly  draws  the  retrograde-moving  rim  inward.  This 
causes  precessional  motion  which  turns  the  rear  rim  inward,  and 
the  combined  motions  bring  about  inversion,  as  shown  by  the 
Montgomery  model.  The  wheel  therein,  with  three  freedoms,  is 
supported  by  gimbals  at  one  end  of  a  horizontal  bar  and  is  bal- 
anced by  a  weight  at  the  other  end.  The  revolution  of  the  bar 
round  a  central  support  causes  the  retrograde-rotating  wheel  to 
invert  its  poles,  which  move  in  spirals  as  the  wheel  turns  over. 

14 


This  motion  represents  the  action  of  the  planets  as  inversion 
proceeds,  and  it  is  not  explained  by  the  theory  of  Professor  Picker- 
ing. It  does  not  occur  when  the  stand  of  a  gyroscope  is  turned 
against  the  rotation  of  the  wheel  while  resting  on  the  same  spot. 
In  such  case  inversion  takes  place  but  in  a  different  manner,  due 
to  the  application  of  a  couple  to  a  stationary  body — an  artificial 
condition  which  does  not  obtain  in  the  case  of  the  planets.  Their 
situation  and  action  are  correctly  represented  by  the  rotating  and 
revolving  wheel  in  the  model. 

The  wheel  and  the  planets  invert  with  a  reeling  motion. 
The  apparent  circles  made  by  the  poles  of  the  earth  in  twenty- 
five  thousand  years,  for  instance,  are  really  spirals  and  accord- 
ingly not  re-entrant.  In  the  case  of  a  planet,  each  particle  of 
matter  is  pulled  by  attraction,  and  in  the  case  of  the  model,  the 
pulling  force  is  exerted  upon  the  center  of  the  wheel  with  pre- 
cisely the  same  result.  In  each  case  it  is  as  if  there  were  a 
contactual  pull  at  the  center.  The  retrograde  and  relatively 
slowly  moving  particles  are  pulled  further  inward  in  a  given 
period  than  the  forward  and  relatively  rapidly  moving  particles. 

And  when  the  wheel  in  the  model  has  turned  over  and  settled 
down  in  a  horizontal  position  it  remains  in  this  position  as  it 
rotates  and  revolves,  because  then  the  inner  parts  tend  to  move 
towards  the  center  of  revolution  and  the  outer  parts  away  from  it. 

But  attraction  is  not  satisfied  until  forward  rotation  of  a  planet 
is  stopped  relatively  to  the  central  body.  This  stoppage  is  due 
to  the  unequal  effects  of  attraction  on  all  the  particles  which  com- 
pose the  planet,  including  the  water  on  its  surface,  and  in  the 
model  is  represented  by  friction  on  the  wheel  corresponding  in 
the  case  of  a  planet  to  the  "friction"  to  which  it  is  subjected  by 
attraction.  But  such  "friction"  is  not  dependent  upon  the 
substance  of  the  planet;  attraction  will  retard  the  rotation  of  a 
tideless  body.  The  rotation  of  a  bar  as  well  as  a  viscous  or  elastic 
planet  will  be  affected  by  attraction.  And  if  rotation  were  stopped 
(absolutely)  attraction  would  start  it  whether  the  body  was  a  bar 
or  any  other  real  body,  regardless  of  tides.  But  the  rotation 
would  never  exceed  the  revolution.  Professor  Newcomb 
("Popular  Astronomy",  p.  315)  says  that  we  have  "every 
reason  to  believe"  that  if  the  rotation  of  the  moon  were  consid- 
erably increased  it  would  never  stop.  But  I  submit  that  he  is 
either  in  error  regarding  this  or  he  is  in  error  in  his  statement 
that  if  the  rotation  were  only  slightly  increased  the  moon  would 
be  drawn  into  line  by  attraction.  His  fly-wheel  on  an  axle  would 

15 


not  stop  in  the  absence  of  friction  but  would  continue  to  oscillate. 
But  his  illustration  was  faulty  because  while  it  is  true  ..that  when 
the  near  end  of  a  rotating  bar  in  space  is  moving  towards  the 
attracting  body,  rotation  will  be  accelerated  and  when  it  is  moving 
away  from  the  attracting  body,  rotation  will  be  retarded,  yet  the 
latter  force  is  longer  in  operation  than  the  former.  It  is  not  as  if 
the  bar  were  turning  on  a  fixed  axle.  The  difference  between  these 
forces  is  the  measure  of  the  retardation  of  rotation.  The  tideless 
moon,  no  matter  how  much  its  rotation  was  increased  or  decreased, 
would  be  brought  back  by  attraction  to  its  present  attitude.  Where 
there  are  tides,  however,  there  is  a  steady  pull  in  the  same  direc- 
tion which  continuously  retards  rotation. 

Let  me  add  that  Professor  Montgomery  asserted  that  the  attrac- 
tion of  the  sun  had  inverted  the  orbit  of  the  moon.  The  retro- 
grade revolution  of  a  satellite  is  unstable  for  the  same  reason 
that  the  retrograde  rotation  of  a  planet  is.  In  the  case  of  retro- 
grade revolution  or  rotation  by  reason  of  the  cancellation,  meas- 
urably, of  forward  motion,  the  outer  body  or  part  is,  as  it  were, 
nearer  the  attracting  body.  If  the  backward  movement  exactly 
equals  the  forward  movement,  the  body  or  part  will  be  (in  rela- 
tion to  the  point  of  attraction)  motionless,  and  if  free  to  do  so, 
will  move  directly  towards  it.  Herein  lies  the  solution  of  the 
problem. 

I  should  appreciate  the  privilege  of  submitting  to  you  here- 
after, by  way  of  additional  argument  in  support  of  Professor 
Montgomery's  views,  some  criticisms  of  the  writings  of  Mr.  Dar- 
win, Professor  Moulton,  and  others  who  have  dealt  with  this  most 
interesting  subject. 

In  re  Pickering. 

In  "Astronomy  and  Astro-physics"  for  June,  1893,  Professor 
William  H.  Pickering  advanced  the  theory  that  a  retrograde  rotat- 
ing planet  would  be  inverted  by  the  action  of  "annual  tides".  In 
September,  1893,  however,  in  the  same  journal,  referring  to  this 
theory,  Professor  Pickering  said: 

"It  does  not,  however,  fully  explain  the  case  of  rapidly 
rotating  bodies  like  the  planets,  since  in  their  case  the  inver- 
sion could  not  proceed  much  further  than  to  make  the  axis 
coincide  with  the  plane  of  the  orbit." 

But  in  all  subsequent  articles,  the  last  appearing  in  the  October, 
1917,  number  of  "Popular  Astronomy",  he  ignores  this  objection 

16 


which  he  himself  urged  against  his  theory,  and  insists  that 
"annual  tides"  will  completely  invert  a  retrograde-rotating  planet. 
While  he  did  not  state  the  reason  why  "annual  tides"  could  not 
cause  inversion  to  "proceed  much  further  than  to  make  the  axis 
coincide  with  the  plane  of  the  orbit",  it  seems  probable  that  what 
he  had  in  mind  was  this:  he  assumed  that  when  the  rotation 
changed  from  retrograde  to  forward  the  tidal  bulge  would  shift. 
According  to  this  view  when  the  rotation  was  retrograde  the  bulge 
would  be  carried  forward  by  rotation  and  be  pulled  back  by  attrac- 
tion, and  when  the  rotation  was  forward  the  bulge  would  be  car- 
ried back  and  be  pulled  forward.  The  backward  pull  would 
precessionally  tilt  the  axis  one  way  and  the  forward  pull  would 
tilt  it  the  opposite  way.  Hence  the  "annual  tide"  would  not 
invert  the  planet  beyond  the  point  where  the  axis  was  brought 
into  coincidence  with  the  orbital  plane.  Assume,  for  instance, 
that  the  equator  of  Uranus,  from  some  cause,  was  tilted  over  to- 
wards the  sun  beyond  90°.  Then  the  tidal  bulge  would  shift  to  the 
other  side  and  attraction  of  it  would  precessionally  tilt  the  equator 
back  to  90°.  But  notwithstanding  this  obvious  objection  to  his 
theory,  which  Professor  Pickering  himself  presumably  saw,  he  has 
strangely  ignored  it  and  has  continued  to  assert  that  the  effect 
of  the  "annual  tide"  would  be  to  make  the  "process  continue 
without  intermission  until  the  two  planes  coincide".  But  if 
"tidal  friction"  were  the  controlling  force,  then  it  is  manifest,  as 
Mr.  Darwin  says,  that  obliquity  in  the  case  of  the  earth  is, 
and  for  a  long  time  has  been,  increasing  ("The  Tides",  p.  310). 
Clearly  if  the  axis  of  the  earth  were  being  precessionally  tilted 
by  "tidal  friction",  it  would  now  be  inclining  into  greater 
obliquity  since  the  tidal  pull  is  against  the  rotation.  But  Darwin, 
as  well  as  Pickering,  was  in  error  because  "tidal  friction"  is  not 
in  control.  The  axis  is  not  precessionally  tilted  by  "tidal  fric- 
tion", but  directly  tilted  by  attraction  on  the  equatorial  bulge  of 
the  planet.  Hence  where,  as  in  the  case  of  the  earth,  the  rotation 
is  forward,  obliquity  must  decrease.  And  where  the  rotation  is 
retrograde,  the  far  side  of  the  planet  (assuming  adequate  rotation) 
is  virtually  nearer  than  the  near  side.  The  outer  retrograde- 
rotating  parts  move,  relatively  to  the  attracting  'body,  slower  than 
the  inner  forward-moving  parts.  They  therefore  are  drawn 
further  inward  in  a  given  time  than  the  relatively  more 
rapidly  moving  inner  parts.  If  the  retrograde  rotation  were 
sufficient  to  cancel  the  forward  motion  of  the  planet,  the  retro- 
grade-moving parts  would  be  at  a  standstill,  relatively  to  the 


17 


attracting  body,  and  if  free  to  do  so  would  move  directly  towards 
it.  The  planet  would  therefore  turn  over.  But  as  it  is  rotating,  the 
tilting  is  accompanied  by  precessional  motion;  the  poles  move  in 
spirals  and  will  continue  to  do  so  until  the  planet  has  been  com- 
pletely inverted  and  rotates  in  a  forward  direction  on  an  axis 
perpendicular  to  the  orbital  plane.  Professor  Pickering's  conclu- 
sion was  correct,  but  he  assigned  the  wrong  cause.  It  is  true,  as 
he  contends,  that  if  a  rotating  planet  without  revolution — assum- 
ing an  ideal  case  for  illustration — were  turned  against  its  rotation, 
its  axis  would  tilt  90°.  And  it  is  also  true  that  if  the  planet  were 
thereafter  subjected  to  the  same  force,  it  would  turn  completely 
over,  as  is  proved  by  rotating  the  stand  of  a  gyroscope  in  a  direc- 
tion, at  first  opposite  to  the  spin  of  the  wheel,  and  when  the  spin 
is  reversed,  with  it.  But  as  above  pointed  out  (and  as  Professor 
Pickering  himself  saw  in  September,  1893)  " tidal  friction",  in 
the  case  of  a  rapidly  rotating  planet,  does  not  work  in  this  fashion. 
It  does  not  tend  to  cause  rotation  in  the  same  direction  through- 
out. Assuming,  as  Professor  Pickering  contends,  that  when  the 
rotation  is  retrograde  it  tends  to  turn  the  planet  against  such 
rotation,  yet  when  the  rotation  becomes  forward,  it  tends  to  turn 
the  planet  in  the  opposite  direction. 

Nor  does  the  action  of  the  gyroscope  give  any  support  to 
Professor  Pickering's  theory.  On  the  contrary  it  demonstrates 
it  to  be  erroneous.  If  the  wheel  in  the  Montgomery  model  is 
placed  in  the  position  of  Uranus  with  the  axis  pointing  towards 
the  central  support  and  be  given  a  retrograde  rotation,  the  retro- 
grade-rotating rim  tilts  inward  when  the  wheel  is  revolved  and  the 
rear  rim  precesses  inward,  which  proves  conclusively  that  the  tilt- 
ing of  the  axis  is  not  a  precessional  effect  of  the  "annual  tide",  as. 
Prbfessor  Pickering  supposes,  but  that  it  is  the  cause  of  a  preces- 
sional movement  opposite  to  the  direct  movement  which  the 
"annual  tide"  would  cause  if  it  were  controlling.  The  inward  tilt- 
ing of  the  retrograde-rotating  rim  is  caused  directly  by  the  pull  at 
the  center  of  the  wheel  which  produces  precisely  the  same  effect 
as  attraction  on  all  the  parts  of  a  retrograde-rotating  planet.  In 
both  cases  the  retrograde  (and  therefore  relatively  slowly)  moving 
parts  yield  to  a  greater  degree  to  the  pull,  applied  in  the  one 
case  at  the  center  and  in  the  other  upon  the  parts,  than  the  rela- 
tively rapidly  moving  forward  parts.  The  planet  and  the  wheel 
turn  over  (the  poles  moving  in  spirals)  until  they  spin  in  a 
forward  direction  on  perpendicular  axes  and  continue  so  to  rotate. 


18 


In  the  foregoing  the  expression  "tidal  friction"  has  been 
employed  merely  as  a  convenient  term.  As  I  have  elsewhere 
shown,  the  friction  of  the  tides  is  a  false  quantity.  Retardation  of 
rotation  is  due  not  to  friction  (which  accompanies  acceleration 
as  well  as  retardation  of  rotation)  but  to  attraction  on  a 
rotating  elongated  body  with  or  without  tides. 

In  re  Moulton. 

In  his  reply  to  Professor  Pickering  ("The  Astrophysical 
Journal",  Vol.  22,  p.  355)  Professor  Moulton  says  that  "the 
question  is  to  determine  the  effects  on  the  rotation  of  a 
planet  of  the  attraction  of  the  sun  for  the  tides  which  it  has 
raised  on  the  planet".  He  confines  the  issue  to  the  effect  on 
a  rotating  but  not  revolving  planet  of  a  "couple"  around 
the  perpendicular  or  Z  axis  of  the  planet.  And  he  contends 
that  such  a  couple  is  incapable  of  inverting  or  even  tilting 
the  planet.  He  says  that  the  "first  principle  of  the  dynamics 
of  rigid  bodies  is  that  all  the  forces  which  act  upon  a  body 
may  be  resolved  into  three  rectangular  components  applied 
to  its  center  of  gravity  and  three  couples  about  three  rectangular 
axes".  He  disregards  the  "three  rectangular  components"  as  not 
involved  in  the  problem,  and  then  proceeds  to  consider  the  effect 
of  one  of  the  couples — that  around  the  Z  axis.  He  justifies  this 
course  because,  he  says,  this  couple  is  the  one  which  Professor 
Pickering  claims  causes  inversion.  He  then  proceeds  to  demon- 
strate that  such  couple  can  only  retard  or  cause  rotation  around 
the  Z  axis — that  it  can  neither  retard  nor  cause  rotation  around 
any  other  axis.  And  he  reaches  the  conclusion  accordingly  that 
the  planet  under  such  an  influence  will  not  invert. 

But  Professor  Moulton 's  remarks  apply  only  to  the  case  of  a 
rigid  perfect  sphere.  If  such  a  sphere  be  rotating  around  the 
Z  axis  and  an  attempt  be  made  to  rotate  it  around  the  X  or  Y 
axis  it  will  not  rotate  around  either,  but  around  a  new  axis. 
The  axis  of  rotation  will  shift  in  the  sphere.  But  this  it  not 
true  of  a  spheroid.  In  such  case  if  the  rotation  be  not  around 
the  Z  axis,  a  couple  around  such  axis  has  the  effect,  not 
of  shifting  the  axis  in  the  body,  but  of  tilting  the  body  itself. 
For  example,  take  a  rotating  planet  whose  axis  lies  in  the  plane 
of  its  orbit  and  points  somewhat  to  one  side  of  the  sun.  It  is 
clear  (eliminating  the  factor  of  translation),  that  attraction  will 
tend  to  turn  the  planet  on  the  Z  axis.  The  effect  thereof  will  not 
be  to  change  the  axis  of  rotation  in  the  planet,  but  to  tilt  the 

19 


planet  itself.  Its  upper  rim  will  move  inward  or  outward,  depend- 
ing upon  which  way  the  planet  is  rotating.  Precession  will  ensue. 
The  gyroscope  furnishes  a  familiar  illustration  of  the  effect  of 
such  a  force.  Hence  Professor  Moulton's  reasoning  and  con- 
clusions, true  only  in  the  case  of  rigid  perfect  spheres,  have 
no  application  to  the  case  of  planets  as  they  are.  It  is  sur- 
prising that  he  should  have  sought  to  apply  to  the  planets  which 
are  not  rigid  spheres,  principles  of  mechanics  inapplicable  to 
them.  After  saying  that  the  question  is  what  effect  tidal  action 
has  on  rotation,  he  proceeds  to  lay  down  principles  relating  to 
" rigid"  bodies,  which  of  course  have  no  tides.  If  a  body  were 
perfectly  rigid  and  spherical,  attraction  would  not  affect  its 
rotation  at  all.  And  if  it  had  no  rotation,  attraction  would  not 
start  rotation.  Hence,  even  in  the  narrow  way  in  which  he 
viewed  the  problem,  Professor  Moulton  was  in  error.  He  was 
also  in  error  in  eliminating  the  factor  of  translation. 
As  I  have  shown,  in  this  lies  the  explanation  of  the  inversion 
of  the  planets.  In  considering  the  effect  of  attraction  on  a 
particle  of  a  planet,  we  should  take  into  account  not  only  its 
rotation  motion  but  its  translation  motion  as  well.  It  is  the  motion 
of  a  particle  in  relation  to  the  attracting  body  which  we 
have  to  consider.  This  proposition  was  overlooked  by 
Professor  Pickering  also.  The  inward  tilting  of  the  retrograde- 
rotating  rim  of  Uranus  is  not  precessional  as  he  supposes — not  an 
effect  of  attempted  rotation  around  the  Z  axis.  It  is  primary. 
It  is  not  caused  by  "annual  tides"  but  directly  by  attraction. 
This  primary  motion  causes  precessional  motion  of  the  rear  rim 
of  the  planet  which  accordingly  turns  inward  as  the  retrograde- 
rotating  rim  tilts  inward.  This  is  proved  by  the  Montgomery 
model. 

In  "Popular  Astronomy"  for  February,  1918,  Professor  Moul- 
ton says:  "It  is  sometimes  supposed  that  a  planet  may  have  orig- 
inally rotated  in  one  direction  and  that  tides  may  have  caused 
it  to  rotate  in  the  opposite.  *  *  *  This  can  have  been  brought 
about  only  by  forces  having  a  moment  about  an  axis  perpendic- 
ular to  the  plane  of  the  orbit  of  the  planet."  Such  a  force  or 
couple  would  of  course  first  stop  the  rotation  and  then  start 
it  in  the  opposite  direction  if  it  were  absolutely  opposite  to  the 
rotation,  but  otherwise  it  would  invert  the  planet.  Proof  of  this 
is  furnished  by  the  gyroscope.  If  the  wheel  be  spun  in  one  direc- 
tion and  the  stand  be  rotated  in  the  opposite  direction  (approxi- 
mately) the  wheel  will  be  inverted.  The  reason  for  this  is  plain. 

20 


The  two  rotations  are  not  absolutely  opposed  and  hence  the  body 
moves  along  the  line  of  least  resistance*— it  turns  over.  But  as  I 
have  pointed  out,  attraction  does  not  work  like  an  artificial 
couple. 

Professor  Moulton  was  not  happy  in  his  exposition  of  the 
mechanics  of  rotation.  While  from  a  purely  theoretical  point  of 
view  it  may  be  unobjectionable  to  say,  as  he  does,  that  "any  rota- 
tion may  be  resolved  into  rotations  around  three  axes",  yet  this 
is  a  practical  impossibility.  While  it  is  true,  of  course,  that  a 
force  which  causes  rotation,  may  be  resolved  into  three  forces 
which  taken  separately,  will  cause  three  different  rotations  and 
when  combined  but  one — the  rotation  around  the  so-called  "in- 
stantaneous" axis — yet  such  single  rotation  is  the  only  real  rota- 
tion. While  in  a  mathematical  sense  it  may  be  regarded  as  the 
"sum"  of  three  rotations,  such  rotations  cannot  actually  co-exist 
in  the  same  sphere.  Nor  will  the  axis  of  rotation  shift  in  the  case 
of  any  real  body.  Assuming  that  there  may  be  a  real  "perfect 
sphere"  yet  the  application  of  a  force  which  causes  rotation  will 
convert  it  into  a  spheroid;  and  hence  when  an  attempt  is  made  to 
rotate  it  on  a  right-angular  axis  instead  of  the  axis  shifting  in 
the  body,  the  body  will  tilt  and  thereby  cause  precession.  Professor 
Moulton 's  failure  to  note  this  accounts  for  the  errors  into  which 
he  fell. 


In  re  Darwin. 

In  "The  Tides"  and  in  his  article  in  the  Encyclopedia  Bri- 
tannica,  Mr.  Darwin  says  that  tidal  friction  is  increasing  the 
earth's  obliquity.  He  says  ("The  Tides",  p.  311)  that  it  is 
"certain  that  a  planet  rotating  primitively  without  obliquity 
would  gradually  become  inclined  to  its  orbit  although  probably 
not  to  so  great  an  extent  as  we  find  in  the  case  of  the  earth". 
And  (p.  310)  that  "at  present  and  for  a  long  time  in  the 
past",  the  earth's  obliquity  has  been  increasing.  If  Professor  Mont- 
gomery's views  are  correct,  Mr.  Darwin  has  here  fallen  into  error. 
His  error  arises  from  ignoring  the  orbital  motion  of  the  earth. 
With  him  there  is  no  "forward"  or  "retrograde"  rotation.  The 
earth  is  rotating  but  not  revolving.  "Tidal  friction"  under  such 
conditions  would,  as  he  says,  precessionally  cause  a  tilting  of  the 
axis  unless  it  was  absolutely  perpendicular  to  the  tidal  force 
— obliquity  would  increase.  But  orbital  motion  brings  in  a  factor 
which  renders  Mr.  Darwin's  deductions  erroneous.  Where  there 


21 


is  orbital  motion  rotation  is  either  "forward"  or  "retrograde". 
If  forward,  obliquity,  as  in  the  case  of  the  earth,  will  decrease; 
and  if  retrograde,  the  planet  will  invert,  thereby  changing 
retrograde  into  forward  rotation,  and  thereupon  obliquity  will 
decrease  until  the  planet  rotates  on  an  axis  perpendicular  to 
the  plane  of  the  orbit. 

But  obliquity  would  not  increase,  Mr.  Darwin  says,  if  there 
were  less  than  two  days  in  the  month — in  such  case  instability 
would  not  exist.  He  does  not  explain  the  reason  for  this  distinc- 
tion, but  it  cannot  be  well-founded.  The  rapidly  moving  outer 
parts  and  the  slowly  moving  inner  parts  (relatively  to  the 
attracting  forces)  will  bring  the  axis  of  the  earth  (the  poles  mov- 
ing in  spirals)  to  a  position  at  right  angles  to  the  orbital  plane 
and  keep  it  there.  Hence,  regardless  of  the  number  of  days  in 
the  month,  the  obliquity  of  the  earth's  axis  will  decrease  and  not 
increase,  as  Mr.  Darwin  contends. 

On  page  309  Mr.  Darwin  says  that  "the  attraction  of  the  moon 
and  sun  on  the  equatorial  protuberance  of  the  earth  causes  the 
earth's  axis  to  move  slowly  and  continuously  with  reference  to 
the  fixed  stars";  and  that  "throughout  this  precessional  move- 
ment the  obliquity  of  the  equator  to  the  ecliptic  remains  con- 
stant". But  I  submit  that  such  precessional  motion  would 
be  impossible  if  the  obliquity  remained  "constant".  It  occurs 
because  attraction  tilts  the  axis.  The  near  and  more  slowly  moving 
inner  parts  are  more  strongly  attracted  than  the  far  and  more 
rapidly  moving  outer  parts.  Hence  the  axis  tilts  and  the  tilting 
causes  precession.  The  compounding  of  the  two  motions  results 
in  a.  spiral  motion  of  the  poles.  As  this  spiral  motion  pro- 
ceeds, obliquity  decreases.  If  the  effect  of  "tidal  friction" 
were  to  precessionally  increase  obliquity,  precession  would  be  at 
right  angles  to  actual  precession.  The  earth  could  not  precess 
as  it  does  and  also  precess  in  a  right-angular  direction.  If  its 
"precession"  is  really  precessional,  it  must  be  due  to  a  tilting  of 
the  axis  in  a  direction  to  decrease  obliquity.  If,  on  the  other 
hand,  the  "precession"  of  the  earth  is  not  really  precessional,  but 
a  primary  motion  caused  by  "tidal  friction",  then  such  motion 
would  precessionally  tilt  the  axis  into  greater  obliquity  as  it 
proceeded.  And  this  would  be  the  case  if  the  earth  had  no  orbital 
motion.  This  is  shown  by  turning  the  stand  of  a  stationary 
gyroscope  against  the  rotation  of  the  wheel.  Mr.  Darwin's  con- 
tention is  based  on  this  proposition.  (See  Ency.  Brit.,  p.  377.) 


22 


But  attraction  on  the  rotating  and  revolving  earth  produces  no 
such  effect.  It  is  true  that  in  such  case  also  rotation  is  re- 
tarded but  greater  obliquity  does  not  ensue  because  the  backward 
and  relatively  slowly  moving  inner  parts  tend  to  go  inward  and 
the  relatively  rapidly  moving  outer  parts  tend  to  go  outward. 
The  planet  is  elongated  towards  and  away  from  the  central 
attracting  body.  This  prevents  increase  of  obliquity.  With  or 
without  tides  .attraction  will  cause  decrease  of  obliquity.  If 
there  were  no  oceans  on  the  earth,  rotation  would  still  be  re- 
tarded, but  at  the  same  time  obliquity  would  continue  to  decrease 
until  the  planet  spun  on  a  perpendicular  axis. 

Mr.  Darwin's  views  find  no  support  in  the  actual  conditions. 
Inversion  is  proportional  to  the  times  of  rotation  and  revolution. 
Mercury  and  Venus  have  probably  made  the  greatest  number  of 
revolutions  and  inversion  in  them  is  complete.  Jupiter  has  the 
most  rapid  rotation  and  inversion  is  almost  complete.  The  least 
number  of  revolutions  has  probably  been  made  by  Neptune  and 
Uranus  and  their  rotations  are  still  retrograde.  Mr.  Darwin 
says  (Ency.  Brit.,  p.  379),  referring  to  Jupiter  and  its  small 
obliquity,  that  its  system  is  "obviously  far  less  advanced 
than  our  own,"  but  that  the  "high  obliquity"  of  Saturn  indicates 
a  later  stage  of  development.  This  is  begging  the  question.  On 
the  other  hand  the  progressive  change  of  inclination  from  Neptune 
to  Jupiter  tends  strongly  to  confirm  the  views  of  Professor  Mont- 
gomery. Nor  is  the  obliquity  of  Mars  and  the  earth  inconsistent 
with  his  views.  This  is  due  to  their  slow  rotation.  And  the 
absence  of  satellites  in  the  case  of  Mercury  and  Venus  is  signi- 
ficant. "Development"  is  most  advanced  in  the  case  of  these 
planets  and  it  seems  not  improbable  that  their  moons  have  been 
wound  up. 


In  re  Newcomb. 

In  my  letter  to  you  of  last  July,  I  disputed  Professor  New- 
comb's  statement  ("Popular  Astronomy",  p.  315)  that  we  have 
"every  reason  to  believe"  that  if  the  moon  were  rapidly  rotated  on 
its  axis  the  earth's  attraction  would  not  retard  such  rotation,  and 
asserted  on  the  contrary  that  however  great  the  moon's  rotation 
might  be,  the  earth's  attraction  would  retard  it  until  it  coincided 
with  its  period  of  revolution.  I  also  asserted  that  if  it  had  no 
rotation  at  all,  the  earth's  attraction  would  start  rotation  and 
increase  it  until  the  period  thereof  equalled  the  period  of  its 


revolution.     But  I  did  not  therein  elaborate  the  reasons  for  these 
conclusions.     I  will  now  do  so. 

Consider  first  the  case  where  the  moon  has  no  rotation.  It  is 
clear  that  the  effect  on  the  moon's  rotation  of  the  earth's  attrac- 
tion is  the  same  as  if  the  earth  revolved  around  the  moon  instead 
of  the  moon  revolving  around  the  earth.  Let  the  moon  be  repre- 
sented by  a  bar  without  rotation  (a-b)  and  the  earth,  (c)  as 
revolving  around  the  moon,  thus : 


The  attraction  of  c  on  b  is  greater  than  on  a,  and  hence  as  c 
revolves  around  a-b  it  will  start  a-b  rotating.  Assume  that  c  has 
revolved  and  a-b  rotated  until  their  positions  are  as  follows : 


Thereafter  c  will  more  strongly  attract  a  than  b — the 
rotation  of  a-b  will  be  retarded.  But  the  period  of  retardation 
will  be  less  than  the  period  of  acceleration,  a-b  does  not,  as 
Newcomb  erroneously  assumed,  rotate  on  a  fixed  axle,  a  as  well 
as  b  is  attracted  by  c.  There  is  no  "couple".  Hence,  although 
a-b  rotates,  a  at  all  times  is  moving  towards  c.  And  as  a  con- 
sequence when  c  reaches  a,  a-b  will  still  be  rotating;  and  on  the 
next  acceleration  period  will  rotate  further  than  on  the  preceding 
acceleration  period.  The  acceleration  will  increase  until  a-b 
rotates  in  the  same  period  that  c  revolves. 


24 


Consider  now  the  case  where  the  rotation  of  a-b  is  more  rapid 
than  the  revolution  of  c.  Assume  that  a-b  will  have  rotated  and 
c  revolved  until  their  positions  are  as  follows: 


Thereafter  c  will  attract  a  more  strongly  than  6,  thereby  accel- 
erating the  rotation  of  a-b.  But,  as  above  explained,  the  period 
of  acceleration  will  be  shorter  than  the  period  of  retarda- 
tion. Hence  when  a  reaches  c  the  rotation  of  a-b  will  be  slower 
than  at  the  start.  Thus  alternately  the  rotation  of  a-b  will  be 
retarded  and  accelerated,  but  the  periods  of  retardation  will  be 
longer  than  the  periods  of  acceleration,  thereby  decreasing  the 
rotation  of  a-b  until  it  coincides  with  the  revolution  of  c. 

Hence  the  question  of  " tides"  in  the  moon  is  immaterial.  Tides 
or  no  tides,  attraction  will  increase  or  decrease  the  period  of  the 
moon's  rotation  until  it  corresponds  with  the  period  of  its  revolu- 
tion. Where  however  there  are  tides,  the  retardation  or  accelera- 
tion is  continuous,  but  where  there  are  no  tides,  rotation  is  alter- 
nately retarded  and  accelerated,  but  the  periods  thereof  are  dif- 
ferent. Where  the  moon  has  no  rotation,  the  periods  of  accelera- 
tion will  be  longer,  and  where  it  has  excessive  rotation  they  will 
be  shorter.  It  is  also  clear  that  the  attraction  of  the  sun  on  the 
moon  cannot  prevent  the  consequences  due  to  the  earth's  attrac- 
tion, which  is  greater  than  that  of  the  sun. 

Hence  Newcomb  was  in  error  in  his  statement  that  the  moon's 
rotation  would  not  be  retarded  if  it  were  several  times  greater 
than  it  is;  and  Ball  erred  in  saying  that  when  the  tides  in  the 
moon  shall  have  subsided,  the  periods  of  rotation  and  revolution 
will  no  longer  coincide;  and  Moulton  erred  in  saying  that  the 
identity  of  the  periods  of  rotation  and  revolution  was  due  to  some 
unconsidered  influence;  and  Darwin  erred  in  saying  that  if 
there  were  "oceans"  on  the  moon,  solar  tides  would  retard  its 
rotation. 

The  foregoing  demonstration  also  disproves  Mr.  Darwin's  con- 
tention that  the  attraction  of  the  earth  is  retarding  the  moon's 
orbital  velocity.  Let  a-b  represent  the  earth  and  c  the  moon.  As 

25 


we  have  seen,  when  the  rotation  of  a-b  is  more  rapid  than  c's 
revolution,  c's  orbital  velocity  will  alternately  be  accelerated  and 
retarded,  but  the  periods  of  acceleration  will  be  longer  than  the 
periods  of  retardation.  And  where  there  are  tides  on  or  in  the 
central  body,  the  orbital  velocity  of  the  revolving  body  will  be 
continuously  accelerated.  Hence  while  the  moon  is  retarding  the 
earth's  rotation,  the  earth  is  accelerating  the  moon's  revolution; 
and  such  retardation  and  acceleration  will  continue  (in  diminish- 
ing degrees)  until  the  period  of  the  earth's  rotation  is  the  same  as 
the  period  of  the  moon's  revolution. 

Letter  of  January  27,  1919,  Accompanying 
"The  Burial  of  a  World." 

Professor  Montgomery  informed  me  that  the  planets  and 
their  satellites  would  finally  be  drawn  by  attraction  into  the  sun. 
He  did  not  however  state  the  reasons  which  led  him  to  this  con- 
clusion. I  have  recently  been  considering  this  subject,  and  in 
another  paper  enclosed  herewith  I  advance  grounds  which  I 
think  sustain  this  proposition. 

In  his  "The  Evolution  of  Worlds",  Professor  Lowell  has  a  chap- 
ter entitled  "The  Death  of  a  World".  My  paper,  which  I  have 
entitled  "The  Burial  of  a  World"  fittingly  supplements  it.  Per- 
haps "Cremation  of  a  World"  would  be  a  better  title.  He 
contends,  just  as  Montgomery  did,  that  a  retrograde-rotating 
planet  is  in  unstable  equilibrium  (p.  140,  et  seq.),  and  that 
forward  rotation  will  be  acquired  by  inversion,  and  that  such 
rotation  will  be  retarded  until  the  periods  of  rotation  and  revolu- 
tion coincide. 

Professor  Lowell  did  not,  however,  comprehend  the  operation 
of  the  forces  by  which  these  effects  are  brought  about.  He  asserts 
that  the  tides  "necessarily  act  as  brakes  upon  the  planet's  spin". 
This  is  not  so.  The  water  on  the  earth  is  not  external  to  it  but  a 
part  of  it.  The  reduction  of  rotation  would  be  the  same  if  the 
water  were  frictionless,  or  if  friction  were  increased  to  the  point 
of  rigidity.  It  is  due  to  the  shape  of  the  earth.  Mr.  Darwin's 
contention  that  the  earth's  rotation  would  not  be  retarded  if  it 
was  composed  of  an  elastic  and  frictionless  substance  is,  I  submit, 
clearly  erroneous.  In  such  case  the  retardation  of  the  whole 
planet  would  be  more  uniform.  Tidal  friction  does  not  retard 
rotation.  It  decreases  the  retardation  of  the  water  (part  of  the 
earth)  and  increases  the  retardation  of  the  underlying  core. 

26 


Energy  is  lost  by  tidal  friction,  but  it  is  not  the  cause  of  the 
reduction  of  the  rotational  momentum  of  the  whole  body.  "Fric- 
tion" is  a  false  quantity.  If  the  rotation  of  the  earth  were  slower 
than  the  moon's  revolution,  it  would  be  accelerated  by  the  moon's 
attraction  and  such  acceleration  would  be  accompanied  by  tidal 
friction.  Retardation  of  rotational  momentum  of  a  planet  is 
caused  by  a  power  which  also  causes  friction  between  its  parts, 
but  such  friction  does  not  reduce  the  rotational  momentum.  It 
merely  dissipates  energy.  The  retardation  of  some  parts  by  con- 
tact with  other  parts  is  offset  by  the  reduction  of  the  retardation 
of  such  other  parts.  The  rate  of  retardation  (or  acceleration)  of 
the  whole  planet  is  the  same  regardless  of  the  incidental  friction 
between  the  parts.  As  the  period  of  rotation  expands  to  the 
period  of  revolution,  with  or  without  tidal  friction,  energy  declines 
because  when  the  periods  of  rotation  and  revolution  are  the  same, 
the  parts  of  the  planet  instead  of  traveling  around  the  sun  in 
wave  lines,  move  in  approximate  circles.  This  was  not  perceived 
by  Professor  Lowell,  although  it  is  necessarily  involved  in  the 
general  principle  which  he  expounded.  It  is  true,  of  course,  that 
tidal  friction  generates  heat,  and  that  the  radiation  of  such  heat 
reduces  energy,  but  in  the  case  of  a  body  without  tides  heat  will 
also  be  generated  by  the  strains  and  distortions  to  which  it  is 
subjected  by  attraction.  In  both  cases  the  generated  and  dissipated 
heat  measures  the  loss  of  energy  resulting  from  the  change  to  cir- 
cularity of  the  orbits  of  the  parts  of  the  planet.  And  where 
rotation  is  caused  by  attraction  heat  will  be  generated  either  by 
tidal  friction  or  distortions  dependent  upon  the  character  of  the 
body;  and  energy  in  such  cases  will  be  dissipated  by  radiation 
of  the  heat.  In  both  instances,  the  one  where  rotation  is  retarded 
and  the  other  where  it  is  initiated  and  accelerated,  energy  will 
be  lost  as  the  orbits  of  the  parts  of  the  planet  become  circular. 

Assuming  that  the  cause  of  retardation  of  rotation  is  tidal 
friction,  then  in  a  case  where  tidal  friction  ceases  before  its  work 
is  accomplished,  ''superfluous  energy",  which  excessive  rotation 
involves,  would  never  be  "dispensed  with".  If  it  be  true,  as 
Professor  Lowell  asserts,  that  Nature  is  ridding  itself  of  "super- 
fluous energy ' '  in  the  solar  system,  then  clearly  it  must  be  employ- 
ing some  agent  other  than  tidal  friction  to  produce  this  result. 

Professor  Lowell  is  also  in  error  in  saying,  as  he  does,  that 
inversion  is  caused  by  "tidal  action",  as  I  have  shown  in  my 
analysis  of  Professor  Pickering's  "annual  tide"  theory. 

27 


He  is  further  in  error  in  asserting  (p.  147)  that  a  planet's 
"tidal  bulges"  tend  to  "carry  over"  its  satellites.  Con- 
sider the  case  of  the  earth,  for  instance.  The  orbit  of  the  moon 
was  not  inverted  by  the  earth,  but  by  the  sun.  The  inversion  of 
the  earth  is  not  the  cause  of  the  "carrying  over"  of  the 
moon,  but  the  effect  (chiefly)  of  the  moon's  influence  as  it 
was  "carried  over"  by  the  sun,  the  differential  attraction,  of  the 
moon  being  greater  than  that  of  the  sun.  The  fact  that  the 
earth's  inversion  has  lagged  behind  the  inversion  of  the  moon's 
orbit,  conclusively  disproves  Professor  Lowell's  theory.  His 
"distrust"  on  this  point  was  well  founded. 

Professor  Lowell  says  that  Sir  Robert  Ball  was  the  first  to  sug- 
gest the  "argument"  he  makes  (p.  145).  Ball's  views  are  set  forth 
in  "The  Earth's  Beginning",  and  in  "The  Story  of  the  Heavens", 
but  that  he  did  not  comprehend  the  nature  and  operation  of  the 
forces  which  were  bringing  about  the  results  which  he  said  were 
impending  is  disclosed  by  his  statement  in  the  latter  work  that 
retardation  is  caused  by  tidal  friction,  and  by  his  further  statement 
that  when  the  tides  in  the  moon  shall  have  subsided,  "there  would 
then  be  no  longer  any  necessary  identity  between  the  period  of 
rotation  and  that  of  revolution ' ' ;  and  hence  that  it  is  possible 
that  future  generations  may  see  the  other  side  of  the  moon.  This 
is  not  so,  as  I  have  shown.  Gravity  would  keep  the  long  diameter 
of  the  moon  pointing  towards  the  earth,  if  the  moon  were 
as  rigid  as  it  is  possible  for  any  real  body  to  be.  And  if  its 
period  of  rotation  were  increased,  gravity  would  decrease  it  until 
it  coincided  with  its  period  of  revolution. 

There  is  a  remarkable  divergence  in  the  views  on  this  subject  of 
Ball,  Darwin,  and  Moulton.  Ball  says  (see  chapter  on  the  tides 
in  "The  Story  of  the  Heavens")  that  when  the  tides  in  the  moon 
subside,  the  earth 's  control  will  cease ;  Darwin  says  that  the  earth 's 
control  would  cease  now,  owing  to  the  sun's  influence,  if  there 
were  tides  on  it;  and  Moulton  says  that  the  situation  is  inexplic- 
able and  implies  the  presence  of  some  unconsidered  influence! 

It  is  strange  that  Professor  Lowell,  whose  "The  Evolution  of 
Worlds"  was  published  in  19:09,  makes  no  reference  to  either 
Pickering  or  Moulton.  Of  course,  if  Lowell  and  Ball  are  right  in 
their  conclusions,  Moulton  is  in  error,  for  in  his  controversy  with 
Pickering,  he  contended  that  attraction  will  not  invert  a  retro- 
grade-rotating orb.  It  is  no  less  remarkable  that  Moulton  in  his 
"An  Introduction  to  Astronomy"  (1916)  makes  no  reference  to 
Lowell  or  Ball. 

28 


The  Burial  of  a  World. 

In  "The  Evolution  of  Worlds"  (p.  140,  et  seq.),  Professor 
Percival  Lowell  contends  that  a  retrograde-rotating  planet  is  in 
unstable  equilibrium  and  that  stable  equilibrium  will  be  attained 
only  after  it  has  been  inverted  and  its  consequent  forward  rotation 
retarded  until  the  period  thereof  coincides  with  its  period  of 
revolution.  He  attributes  this  result  to  the  dissipation  by 
"Nature"  of  "superfluous  energy"  in  the  solar  system,  and  its 
reduction  to  the  smallest  quantity  consistent  with  the  fixed  amount 
of  moment  of  momentum. 

But  "superfluous  energy"  would  exist  under  the  conditions 
depicted  by  Lowell  in  his  chapter  entitled  "The  Death  of  a 
World".  If,  as  he  argues,  Nature  "abhors"  superfluous  energy, 
it  may  dispense  with  some  of  it  by  contracting  the  orbits  of  the 
planets  until  they  are  drawn  into  the  sun.  It  may,  I  think,  be 
demonstrated  that  the  forces  which  cause  rotation  and  revolution 
will  bring  about  this  result,  as  well  as  inversion  of  retrograde 
rotation  and  retardation  of  forward  rotation,  as  I  have  already 
shown. 

Consider  a  planet  as  not  rotating  or  revolving,  and  as  being 
subjected  only  to  the  attraction  of  the  sun.  Eliminating  for  the 
moment  the  motion  of  the  planet  towards  the  sun,  the  effect  of 
such  attraction  will  be  elongation  of  the  planet  to  the  point  of 
balance  between  the  force  tending  to  elongate  it  and  the  forces 
which  tend  to  maintain  a  spherical  shape.  Now  give  full  effect 
to  the  force  of  gravity  exerted  by  the  sun  on  the  planet,  and  at  the 
same  time  apply  to  the  system  such  tangential  and  rotatory  forces 
that  the  planet  will  rotate  and  revolve  in  the  same  period  and 
in  the  same  direction  as  the  sun  rotates.  Clearly  the  forces  caus- 
ing rotation  and  revolution  of  the  planet  will  differently  affect  its 
parts.  The  parts  nearest  the  sun  will  tend  to  move  towards  it, 
and  the  remote  parts  away  from  it,  the  planet  as  a  whole  moving 
in  an  orbit  which  lies  between  the  lines  of  such  tendencies.  The 
application  of  such  forces  will  increase  the  initial  elongation  of 
the  planet  due  to  gravity  alone,  and  at  the  same  time  slightly 
flatten  it,  as  well  as  the  sun.  As  these  results  ensue  the  force  of 
gravity  on  the  planet  as  a  whole  increases.  The  further  the  sun 
and  planet  depart  from  a  spherical  shape  the  greater  will  be  the 
attraction  of  the  sun.  The  concentrated  point  of  attraction  near 
the  center  of  the  planet  will  shift  towards  the  sun.  And  the  sun 
also  will  be  further  elongated  towards  the  planet.  The  result  is 
that  the  planet,  being  subjected  to  constantly  increasing  attraction, 

29 


winds  inward,  just  as  it  would  do  if  the  mass  of  the  sun  were 
gradually  increased.  As  it  winds  inward  the  period  of  its  rotation 
and  revolution  and  the  period  of  the  sun's  rotation  decrease— 
angular  speed  of  the  system  increases  (the  consequent  flattening 
of  the  sun  and  planet  further  augmenting  their  attraction)  — 
while  the  linear  speed  of  the  planet  decreases.  Thereby  both  the 
orbital  energy  and  moment  of  momentum  of  the  planet  are  reduced. 
The  superfluous  energy  is  " dispensed  with" — by  dissipation  of 
heat  generated  in  both  sun  and  planet — while  the  lost  moment  of 
momentum  is  gained  by  the  sun. 

The  forces  exerted  upon  the  planets  are  continually  altering 
their  shapes.  A  perfect  sphere  exerts  and  has  exerted  upon  it 
for  a  given  mass  a  minimum  force.  As  it  flattens  and  elongates 
its  attraction  progressively  increases.  The  planets  are  falling 
into  the  sun  by  circuitous  routes.  "Like  drops  of  rain:  however 
they  may  linger  in  the  lakes  and  pools,  must  sometime  reach  the 
sea." 

I  am  aware  that  these  views  are  not  in  harmony  with  the 
opinion  entertained  by  Mr.  Darwin,  but  I  think  his  position  is 
clearly  untenable. 

He  contends  ( and  is  supported  in  this  contention  by  Lord  Kelvin 
and  Professor  Moulton)  that  tidal  action  is  retarding  both  the 
rotation  of  the  earth  and  the  revolution  of  the  moon,  but  that  the 
rate  of  retardation  of  the  rotation  is  greater  than  that  of  the 
revolution,  and  hence  that  they  will  ultimately  coincide;  and, 
Mr.  Darwin  asserts,  when  this  occurs  the  earth  and  moon  will 
be  in  stable  equilibrium. 

I  submit  on  the  contrary,  that  the  revolution  of  the  moon  is 
accelerated  by  tidal  action  and  the  rotation  of  the  earth  retarded, 
and  that  such  acceleration  and  retardation  will  last,  in  diminishing 
degrees,  until  the  periods  thereof  are  the  same.  Mr.  Darwin  and 
Lord  Kelvin  ("Natural  Philosophy",  Vol.  1,  p.  255)  admit  that 
"in  the  first  place"  the  moon's  revolution  will  be  accelerated,  but 
assert  that  thereafter  it  will  be  retarded.  But  there  is  no  "in  the 
first  place".  If  it  be  accelerated  "in  the  first  place"  it  is  acceler- 
ated all  the  time.  Mr.  Darwin  concedes  that  his  position  is 
"paradoxical".  In  my  opinion  it  is  more  than  "paradoxical".  It 
amounts  to  saying  that  the  moon's  revolution  is  both  accelerated 
and  retarded  by  the  same  force  at  the  same  time! 

Mr.  Darwin  and  Professor  Moulton  say  that  if  at  the  time  of  the 
"first  identity"  the  rotation  of  the  earth  were  infinitesimally 

30 


slower  than  the  revolution  of  the  moon,  both  periods  would  de- 
crease, but  that  the  period  of  the  revolution  would  decrease  more 
rapidly  than  the  period  of  rotation.  I  contend  on  the  contrary, 
that  in  such  case  the  revolution  of  the  moon  would  be  retarded 
as  the  rotation  of  the  earth  was  accelerated,  and  hence  that  the 
inequality  would  be  instantly  corrected.  Inspection  of  Mr. 
Darwin's  diagram  ("The  Tides",  p.  266)  makes  it  obvious, 
I  submit,  that  the  moon's  revolution  is  accelerated  as  the  earth's 
rotation  is  retarded;  that  is  to  say  that  it  is  more  rapid  than 
it  would  be  if  the  earth  were  perfectly  spherical.  The  acceler- 
ation, however,  will  decline  as  the  rotation  is  retarded.  During  this 
period  the  revolution  of  the  moon  will  be  above  normal  and  the 
revolution  of  the  earth  (with  respect  to  the  moon)  below  normal. 
The  earth  will  gain  the  moment  of  momentum  which  the  moon 
loses,  and  finally  they  will  rotate  and  revolve  as  if  they  were 
united  by  a  bar  extending  through  them.  They  will  thereafter 
slide  along  the  bar  towards  each  other,  under  constantly  increasing 
attraction  and  increasing  angular  velocity,  until  they  coalesce. 
During  this  time  and  thereafter  the  moment  of  momentum  of  the 
system  will  remain  the  same. 

The  asserted  analogy  between  retardation  by  acceleration,  and 
acceleration  occasioned  by  a  resisting  medium,  is  false.  In 
the  latter  case  the  medium  does  not  cause  acceleration;  it  causes 
retardation.  Acceleration  is  caused  by  attraction. 

As  an  indication  of  the  looseness  of  Mr.  Darwin 's  grasp  upon  the 
subject,  I  cite  his  statement  (Ency.  Brit.,  p.  375)  that  the  moon's 
orbital  moment  of  momentum  must  increase  because  that  of  the 
earth's  rotation  diminishes.  This  is  not  so.  It  is  the  retardation 
of  the  earth's  revolution,  with  respect  to  the  moon,  that  requires 
increase  of  the  moon's  orbital  moment  of  momentum.  Ketarda- 
tion  of  rotation  merely  dissipates  energy.  And  acceleration  of 
the  moon's  revolution  would,  of  course,  increase  its  orbital  moment 
of  momentum.  Hence  the  assumed  necessity  for  an  expanded  orbit 
disappears. 

Mr.  Darwin  says  ("The  Tides",  p.  356)  that  "the  general 
question  of  the  limiting  proximity  of  a  liquid  planet  and  sat- 
ellite which  just  insures  stability  is  not  yet  solved".  It  never 
will  be  in  my  opinion.  There  is  no  stability  short  of  concentra- 
tion in  the  sun  of  all  of  the  matter  in  the  planets  and  their 
satellites,  when,  as  above  stated,  a  condition  will  have  been 
attained  where  there  will  be  a  minimum  quantity  of  energy  for  the 

31 


fixed  amount  of  moment  of  momentum  in  the  system.  Perpetual 
revolution  is  only  theoretically  possible — only  where  the  bodies 
are  weighted  points  or  perfect  spheres. 

Further  Comments  on  Mr.  Darwin's  Views. 

Mr.  Darwin  leaves  it  uncertain  as  to  whether  or  not  'he  con- 
siders that  "tidal  friction"  operates  as  a  "brake"  on  the 
earth's  rotation.  The  illustrations  and  statement  with  which  he  be- 
gins ("The  Tides",  pp.  264-5)  indicate  that  he  takes  the  view  that 
it  does,  but  when  his  subsequent  explanation  of  the  operation 
and  effect  of  the  tides  is  examined  closely,  it  appears  that  his 
opinion  was  that  the  real  function  of  "tidal  friction"  is  to 
carry  the  tidal  "protuberance"  forward  so  that  it  will  give 
occasion  for  attraction  to  retard  the  rotation  of  the  whole  earth, 
just  as  if  it  were  an  elongated  rigid  body  placed  obliquely  to 
the  line  of  attraction.  Of  course  the  core  of  the  earth  would 
be  retarded  by  the  friction  between  it  and  the  overlying  waters, 
as  the  waters,  while  being  carried  forward  by  rotation  are 
pulled  back  by  attraction,  but  the  greater  the  friction  the  less 
the  waters  (part  of  the  earth)  would  be  retarded.  Hence  the 
friction  between  the  waters  and  the  core  is  a  false  quantity. 
The  loss  equals  the  gain.  And  as  Mr.  Darwin  himself  pointed 
out,  when  the  waters  are  carried  by  rotation  past  their 
"proper  place"  and  then  pulled  back  by  the  force  of  attrac- 
tion, the  effect  on  rotation  is  the  same  as  if  the  waters, 
so  situated,  were  rigidly  attached  to  the  earth.  Hence  retarda- 
tion will  take  place  without  tidal  friction. 

But  Mr.  Darwin  entertained  the  view  that  if  the  waters  were 
not  carried  past  their  "proper  place",  there  would  be  no  retarda- 
tion of  rotation.  This,  however,  is  clearly  not  so.  Retardation 
would  occur  just  the  same  if  the  waters  were  frictionless  or 
the  earth  perfectly  elastic.  Retardation  is  not  dependent  on 
friction  but  on  the  shape  of  the  earth.  The  less  the  friction 
the  greater  the  response  of  the  waters  to  the  force  of  attraction, 
and  the  elongated  core  also  responds.  Where  there  is  friction 
there  is  less  retardation  of  water  and  more  of  core,  and  without 
friction  more  of  water  and  less  of  core.  It  is  surprising  that 
Mr.  Darwin  overlooked  this  simple  proposition.  Tidal  friction 
no  more  retards  the  earth 's  rotation  than  it  would  be  retarded 
by  plowing  a  furrow  around  it  from  east  to  west. 

32 


Mr.  Darwin  says  ("The  Tides",  p.  288)  that  if  when  the  period 
of  the  "second  identity"  is  reached  there  are  still  "oceans"  on 
the  earth,  its  rotation  will  be  retarded  by  solar  tides.  This  posi- 
tion, I  submit,  is  clearly  untenable.  The  rotation  of  the  earth 
would  still  be  controlled  by  the  moon,  not  the  sun,  whose  dif- 
ferential attraction  is  less.  And  again  on  page  289,  he  says  that 
if  there  were  "oceans"  on  the  moon  solar  tides  would  cause  it 
to  rotate  slower  than  it  revolves.  If  this  were  true,  then  the 
moon  would  now  be  rotating  slower  than  it  revolves.  The  pres- 
ence of  water  on  it  would  not  alter  the  forces  and  laws  which 
control  its  rotation.  And  incidentally  it  may  be  noted  that  this 
view  does  not  harmonize  with  Ball's  contention  that  the  identity 
of  the  moon's  periods  of  rotation  and  revolution  is  due  to  the 
tides  in  it.  Why  does  not  the  sun  work  on  these  tides  and  now 
retard  the  moon's  rotation?  This  "difficulty"  caused  Professor 
Moulton  to  remark  ("An  Introduction  to  Astronomy",  p.  458)  : 

"It  seems  probable  from  this  line  of  thought  that  some  in- 
fluence, so  far  not  considered,  has  caused  the  moon  always  to 
present  the  same  face  to  the  earth. ' ' 

The  "difficulty",  however,  is  removed  by  a  right  understanding 
of  the  problem.  Tides  or  no  tides,  the  rotation  of  the  moon  is 
controlled  by  the  earth,  not  the  sun. 

Mr.  Darwin  assumes  that  at  the  time  the  period  of  the 
moon's  rotation  became  coincident  with  the  period  of  its  revolution 
there  were  still  "oceans"  on  it.  But  such  assumption  has  no 
basis  in  fact.  May  not  the  waters  on  the  moon  have  dried  up 
while  the  moon  still  had  superfluous  rotation?  And  if  so,  how 
was  such  rotation  dissipated?  On  the  tidal  theory  it  never  would 
be,  and  yet  the  strains  and  distortions  to  which  the  rotating  moon 
would  in  such  case  be  subjected  would  of  course  generate  heat 
and  cause  dissipation  of  energy.  This  consideration  confirms  the 
conclusion  that  the  retardation  of  the  rotation  of  a  planet  or 
satellite  is  due  to  its  shape,  as  I  have  demonstrated  in  "In  re 
Newcomb",  and  not  to  tidal  friction.  Attraction  would  "dispense 
with"  the  superfluous  rotation  of  a  crow-bar.  It  is  remarkable 
that  Mr.  Darwin  failed  to  see  this  as  he  saw  (p.  279)  that  tidal 
friction  would  accompany  acceleration  of  rotation,  which  would 
occur  if  the  moon  revolved  (as  in  the  case  of  Phobos)  more 
rapidly  than  the  earth  rotated. 

33 


Mr.  Darwin  ("The  Tides",  p.  338)  quotes  Mr.  Becker  as 
saying : 

"Laplace  assigns  no  cause  for  the  heat  which  he  ascribes  to 
his  nebula.  Lord  Kelvin  goes  further  back  and  supposes  a 
cold  nebula  consisting  of  separate  atoms  or  meteoric  stones 
initially  possessed  of  a  resultant  moment  of  momentum  equal 
or  superior  to  that  of  the  solar  system.  Collision  at  the  center 
will  reduce  them  to  a  vapor  which  then  expanding  far  beyond 
Neptune's  orbit  will  give  a  nebula  such  as  Laplace  postulates." 

But  such  a  theory  is  manifestly  unsound.  A  system  of  revolv- 
ing meteoric  stones  would  not  generate  energy.  It  would 
dissipate  it  and  the  dissipation  would  continue  until  all  the 
stones  were  collected  in  one  cold  rotating  body.  Some  influence 
external  to  the  system  would  be  necessary  to  produce  such  a 
nebula  as  Laplace  postulated. 


34 


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